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Expectation and Variance

The expectation of a random variable, E(X)E(X)E(X)
, is the average of possible values weighted by their probabilities. Formally, it can be defined in two ways:
1.Domain definition: E(X)=∑ω∈ΩX(ω)P(ω)E(X) = \sum_{\omega \in \Omega} X(\omega) P(\omega)E(X)=∑ω∈Ω​X(ω)P(ω)
.
2.Range definition: E(X)=∑xxP(X=x)E(X) = \sum_x x P(X = x)E(X)=∑x​xP(X=x)
.
Expectation has nice properties of linearity: E(X+Y)=E(X)+E(Y)E(X + Y) = E(X) + E(Y)E(X+Y)=E(X)+E(Y)
 and E(aX+b)=aE(x)+bE(aX + b) = aE(x) + bE(aX+b)=aE(x)+b
.
​http://prob140.org/textbook/content/Chapter_08/01_Definition.html​

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